Design of observer-based event-driven controllers for a class of state-dependent nonlinear systems

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Design of observer-based event-driven controllers for a class of state-dependent nonlinear systems Jian Fenga,n, Ning Lia,b a

b

College of Information Science and Engineering, Northeastern University, Shen Yang, China Digital Culture and New Media Technology Research Center of Century College, BUPT, Beijing, China Received 20 May 2015; received in revised form 20 November 2015; accepted 11 February 2016

Abstract The design problems of the observer-based event-driven controllers are investigated for the statedependent nonlinear systems in this paper. An event-driven criterion is proposed to determine whether the newly sampled states of the designed state-dependent observer should be sent out to the controller. As a result, the communication resources can be saved significantly while the burden of the network communication can be reduced. Influenced by the event-driven controller, the closed-loop system is rewritten as the delayed system. The state-dependent integral function is introduced to be the Lyapunov function candidate to obtain less conservative asymptotic stability conditions and preserve the desired H 1 performance for the closed-loop system. The observer gain matrix, the controller gain matrix and the eventdriven parameters are co-designed and co-obtained in terms of solution to a set of linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed method in this paper is illustrated by the numerical examples and the tunnel diode circuit systems. & 2016 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction In practical systems, state-dependent nonlinearities often appear due to unavoidable parameter variations, disturbances, and so on. The class of state-dependent nonlinear systems covers a wider class of nonlinear systems and exists widespread in actual systems, such as the tunnel diode circuit [1,2], the inverted pendulum [3], and the truck trailer [3]. So it is of practical n

Corresponding author. E-mail address: [email protected] (J. Feng).

http://dx.doi.org/10.1016/j.jfranklin.2016.02.012 0016-0032/& 2016 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

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J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

significance to research on the stability analysis and controller synthesis for the state-dependent nonlinear system. In the field of control, it is well known that state feedback control is one of the most commonly methods of control. But because of the difficulty of measurement or the limitation of measuring equipment on the economy and usability in the actual systems, it is difficult for state feedback control. To the best of authors knowledge, state observer is an effective way to estimate the states of the system to complete state feedback control. The classic state observer has been designed and proved to be an efficient tool for state estimation in [4]. The state feedback controller based on the state-dependent observer is designed for the statedependent nonlinear system in this paper. In order to prove that the closed system with the designed controller is asymptotically stable, the integral function [5] with states and estimated states is introduced to be the Lyapunov function. In actual industrial systems, system components are connected through common digital communication network medium. Because of the limited network bandwidth, some uncertainties often happen [6–12], which bring greatly impacts on systems, such as performance degradation, instability, and failures of control. To counteract or to relieve the impacts on systems, event generators are placed into systems to reduce the occupancy bandwidth of network. Compared with periodic sampling mechanism, event-driven mechanism can reduce the release time of the sensor. As a result, communication resources can be saved. In recent years, event-driven methods have been paid more attention [13–26]. For the continuous-time linear system, the observer-based event-driven control problem has been proposed in [24], both continuous- and discrete-time event generators have been designed. Notice that send-on-delta sampling [27–30] allows reduction of event rates and better resource utilization in the networked control systems. But when the system is quasi-stationary, the number of events is quite high by send-on-delta approached (without taking into account noise effects). In this paper, the event-driven criterion is given in the form of energy, which is described by relative increment ratio of energy between the energy increment at current time and the energy at last release time. Only when the ratio is over the threshold, the state is sent out to the controller. Consequently, the burden of the network communication is reduced and the communication bandwidth is saved. To the best of our knowledge, little work has been considered on observer-based event-driven control for the state-dependent nonlinear systems. It motivates authors to research on this issue. In this paper, the event-driven state feedback controller and the robust event-driven state feedback controller are designed for the state-dependent nonlinear systems, respectively. Assume that the system can be controlled through a network medium and only network transmission delays can be considered. The main works of this paper are highlighted as follows: 1. For the state-dependent nonlinear system, based on the designed state-dependent observer, the event-driven criterion is proposed in the form of energy, which can be described by relative increment ratio of energy between the energy increment at current time and the energy at last release time. As a result, the transmission frequency of sampling signals can be significantly reduced, and the burden of the network communication can be reduced. 2. The closed-loop system with the observer and the controller is modeled as the time-varying delayed system. The state-dependent integral function is chosen to be the Lyapunov function candidate to prove that the system is asymptotically stable with the desired H 1 performance. 3. The observer gain, the controller gain and the event-driven parameters are co-designed and co-obtained in terms of solution to a set of LMIs. At last, the effectiveness of the proposed method is illustrated by the numerical examples and the tunnel diode circuit systems. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

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2. Problem description and preliminaries 2.1. Model description Consider the state-dependent nonlinear system, x_ ðtÞ ¼ AðsðxðtÞÞÞxðtÞ þ BðsðxðtÞÞÞuðtÞ; zðtÞ ¼ CxðtÞ; n

ð1Þ q

q

where xðtÞA R , uðtÞA R and zðtÞA R are the system state vector, the control input vector and m the output vector, respectively. sðxðtÞÞ ¼ ½si ðxi ðtÞÞ nonlinear Pmm1 A Ri is the state-dependent i i i vector function of x ðtÞ, and si ðx ðtÞÞ satisfies i ¼ 1 si ðx ðtÞÞ ¼ 1. x ðtÞ is the vector, whose elements belong and BðsðxðtÞÞÞ can be rewritten as: P to x(t).i The system matrices AðsðxðtÞÞÞ Pm i AðsðxðtÞÞÞ ¼ m i ¼ 1 si ðx ðtÞÞAi and BðsðxðtÞÞÞ ¼ i ¼ 1 si ðx ðtÞÞBi , where Ai, Bi, i ¼ 1; 2; …; m, and C are constant matrices of appropriate dimensions. The system (1) can be expressed as, x_ ðtÞ ¼

m X

si ðxi ðtÞÞðAi xðtÞ þ Bi uðtÞÞ:

ð2Þ

i¼1

The system (1) or (2) is a local linear representation of an underlying system. So it can be described by a set of if-then rules that the premise variable is the state of the system. Assume that the system is controlled through a network medium. The sensor is clock-driven, and the sampling period is h40. The state observer is designed such that the estimated state x^ ðtÞ can satisfy lim ðxðtÞ x^ ðtÞÞ ¼ 0. t-1

Assumption 1. The pairs ðAi ; CÞ, i ¼ 1; 2; …; m, are completely observable.

Under Assumption 1, a nature choice of the observer is x^_ ðtÞ ¼

m X

si ðxi ðtÞÞðAi x^ ðtÞ þ Bi uðtÞÞ þ LðzðtÞ z^ ðtÞÞ;

i¼1

z^ ðtÞ ¼ Cx^ ðtÞ; where x^ ðtÞA Rn and z^ ðtÞA Rq are the estimated state vector and the output vector of the observer, respectively. L A Rnq is the gain matrix to be determined. However, the observer is not implementable because it depends on the immeasurable state x(t). Then the following observer where the nonlinear function sðÞ is a function of the estimated state x^ ðtÞ is designed as follows: x^_ ðtÞ ¼

m X

si ð^x i ðtÞÞðAi x^ ðtÞ þ Bi uðtÞÞ þ LðzðtÞ z^ ðtÞÞ;

i¼1

z^ ðtÞ ¼ Cx^ ðtÞ:

ð3Þ

The observer Eq. (3) is feasible, but it is difficult to design the controller to make sure that the observer error system between (2) and (3) is asymptotically stable, as shown in [31] and [32]. Because the independent variables of the state-dependent nonlinear function sðÞ in Eqs. (2) and (3) are different variables x(t) and x^ ðtÞ, respectively. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

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Denoting eðtÞ ¼ xðtÞ x^ ðtÞ, the observer error system is given as follows: ! ! m m m X X X i i i si ðx ðtÞÞAi  LC eðtÞ þ si ðx ðtÞÞAi  si ð^x ðtÞÞAi x^ ðtÞ e_ ðtÞ ¼ i¼1

þ

i¼1

m X

si ðxi ðtÞÞBi 

i¼1

m X

i¼1

!

si ð^x i ðtÞÞBi uðtÞ:

i¼1

Defining ξT ðtÞ ¼ ½eT ðtÞ x^ T ðtÞ, the augmented system is established,  m  X si ðxi ðtÞÞ  si ð^x i ðtÞÞ _ξðtÞ ¼ ðA i ξðtÞ þ B i uðtÞÞ; 0 si ð^x i ðtÞÞ i¼1 where

"

Ai Ai ¼ LC

# Ai ; Ai

ð4Þ

"

# Bi Bi ¼ : Bi

The purpose of this paper is to design an observer-state feedback controller described as uðtÞ ¼ K x^ ðtÞ to make the closed-loop system is asymptotically stable, where K A Rpn is the gain matrix to be determined later. The closed system can be described as,  m  X si ðxi ðtÞÞ  si ð^x i ðtÞÞ _ ¼ ð5Þ ξðtÞ ðA i ξðtÞ þ B i K x^ ðtÞÞ: 0 si ð^x i ðtÞÞ i¼1

2.2. Event driven criterion The event generator is positioned between the sensor and the controller in the networked control system, as shown in Fig. 1, to determine whether the observer state signals should be sent Actuator

Plant

State Observer

u(t )

y (t )

xˆ(t )

Sampler xˆ(kh)

ZOH

Event Generator xˆ(tk h)

u(tk h)

Controller

Fig. 1. Structure of an observer-based event-driven network control system. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

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out to the controller by the following criterion: ½^x ððk þ jÞhÞ x^ ðkhÞT Ω½^x ððk þ jÞhÞ x^ ðkhÞ r ρ^x T ðkhÞΩ^x ðkhÞ;

ð6Þ

where Ω is a positive matrix, ρ A ½0; 1Þ is a constant, j ¼ 1; 2; …. Remark 1. From the event-driven criterion (6), it is possible to note that only some of the sampled states that exceed the threshold ρ can be sent out to the controller. The matrix Ω is the gain matrix of the event-driven criterion (6). The matrix Ω can make the difference values between the two sides of criterion (6) more apparent, such that it is easy to determine whether the state signals should be sent out. Compared with the time-driven method, the network communication bandwidth is saved. In particularly, when ρ ¼ 0, the event-driven criterion (6) turns to the time-driven criterion. Under the action of the event generator, assuming that the release times are t 0 h, t 1 h, t 2 h, …, where t 0 ¼ 0 is the initial time, the release periods are si h ¼ t iþ1 h  t i h. In the process of network transmission, the sensor-controller-actuator transmission delays only be considered, which are time-varying but bounded. Suppose that the transmission delays are τk A ½0; τÞ, where τ ¼ maxfτk g is a positive real constant. As shown in Fig. 2, the released states xðt 0 hÞ, xðt 1 hÞ, xðt 2 hÞ, …, arrive at the actuator at t 0 h þ τ0 , t 1 h þ τ1 , t 2 h þ τ2 , …, respectively. For t A ½t k h þ τk ; t kþ1 h þ τkþ1 Þ, defining τðtÞ ¼ t  t k h and ek ðt  τðtÞÞ ¼ x^ ðt  τðtÞÞ  x^ ðt k hÞ, so τðtÞA ½τk ; ðt kþ1 h t k hÞ þ τkþ1 ÞD½τk ; h þ τ. The event-driven criterion (6) can be rewritten as: for t A ½t k h þ τk ; t kþ1 h þ τkþ1 Þ, eTk ðt  τðtÞÞΩek ðt  τðtÞÞ r ρð^x ðt  τðtÞÞ  ek ðt  τðtÞÞÞT Ωð^x ðt  τðtÞÞ  ek ðt  τðtÞÞÞ:

ð7Þ

The closed system (5) with the observer-based event-driven controller uðtÞ ¼ Kð^x ðt  τðtÞÞ  ek ðt  τðtÞÞÞ can be rewritten as,  m  X si ðxi ðtÞÞ  si ð^x i ðtÞÞ _ ¼ ð8Þ ξðtÞ ðA i ξðtÞ þ B i Kð^x ðt  τðtÞÞ  ek ðt  τðtÞÞÞÞ: 0 si ð^x i ðtÞÞ i¼1 2.3. A Lyapunov function candidate Definition 1 (Khalil [33]). Considering the autonomous system x_ ðtÞ ¼ f ðxÞ, its equilibrium is at the origin xn ¼ 0. Let VðxÞ : Rn -R be a continuously differentiable function, if V(x) satisfies the following conditions: (1) Vð0Þ ¼ 0; (2) for xa 0, VðxÞ40; (3) J xJ -1 ) VðxÞ-1; then V(x) is a Lyapunov function candidate.

Fig. 2. Examples of the different cases of time delays. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

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Consider a line-integral function [5], Z m X si ðψ i ÞP i ψ  dψ; VðxðtÞÞ ¼ 2

ð9Þ

Γð0;xÞ i ¼ 1

where Γð0; xÞ is a path from the origin 0 to the current state x, ψ is a dummy vector for the integral, ðÞ denotes an inner product, P i is a positive matrix satisfying P i ¼ P þ Pi , 3 3 2 i 2 0 p12 ⋯ p1n 0 0 p11 0 7 7 6 6 6 0 pi22 ⋯ 0 7 6 p12 0 ⋯ p2n 7 7: 7 6 6 ð10Þ ; Pi ¼ 6 P¼6 ⋮ ⋱ ⋮ 7 ⋮ ⋱ ⋮ 7 5 5 4 ⋮ 4 ⋮ p1n p2n ⋯ 0 0 0 ⋯ pinn Remark 2. For P i , i¼ 1,…,m, off-diagonal elements all equal to P. According to Lemma 1 in [5], VðxðtÞÞ is to be path-independent, and it can be a Lyapunov function candidate for the system (1) with uðtÞ ¼ 0. Particularly, when i a j, P i ¼ P j , VðxðtÞÞ degrades into the conventional quadratic Lyapunov function.

2.4. Lemmas Lemma 1 (Zhang et al. [6]). For any vector function x(t), matrix P40 and function τðtÞ satisfying 0oτðtÞr τ, the following inequality holds: Z t Z t Z t T x ðsÞ ds P xðsÞ dsr τðtÞ xT ðsÞPxðsÞ ds: t  τðtÞ

t  τðtÞ

t  τðtÞ

Lemma 2 (Zhang et al. [6]). The following inequalities W þ αN 1 o0; W þ αN 2 o0; are equivalent to the following condition W þ βN 1 þ ðα  βÞN 2 o0; where W, N1, and N2 are constant matrices with appropriate dimensions, α40 and β A ½0; α. Lemma 3 (Xiong and James [34]). For matrices R40, X and any scalar γ, the inequality  XR  1 X r γ 2 R  2γX holds.

3. Main results 3.1. Event-driven controller design In this section, an event-driven controller is designed such that the system (8) is asymptotically stable, for t A½t k h þ τk ; t kþ1 h þ τkþ1 Þ. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

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Theorem 1. Considering the closed-loop system (8), for given constants ρ, ρj, and matrix Ω, if there exists matrix K, matrix L, and matrices X i 40, Ri 40, Qi 40, i ¼ 1; 2; …; m, such that for any i; j ¼ 1; 2; …; m and s ¼ 1,2, (11) and (12) hold, then the system (8) with the controller u ¼ Kð^x ðt  τðtÞÞ  ek ðt  τðtÞÞÞ is asymptotically stable: 2 3 T Ψ T2j X j Ψ T4ij Φ1ij þ Q1j 6 7 6 n τM 2 ðρ2j Qj  2ρj IÞ 0 0 7 6 7o0; ð11Þ 6 7 n n I 0 5 4 n n n I 2 3 T Φ1ij þ Q2j Ψ T2j X j Ψ T4ij 6 7 6 n τM 2 ðρ2j Qj  2ρj IÞ 0 0 7 6 7o0; ð12Þ 6 7 n n I 0 5 4 n n n I where

2

3 0 6 7 0 7 6 n 7; Φ11ij ¼ symfX j A ij g þ I T Rj I 1 ; Φ1ij ¼ 6 1 6 n 7 n ðρ 1ÞΩ 0 4 5 n n n  Rj h i h i X j ¼ X j 0 0 0 ; Ψ 2j ¼ ½LC Aj  Bj K  Bj K 0 ; " # " #   Ai Ai  Aj Bi  Bj 0 ; A ij ¼ ; B ij ¼ ; I1 ¼ 0 Ai Bi I   h i  LC 0 L¼ ; Ψ 4ij ¼ L B ij K  B ij K 0 ; LC 0 Φ11ij

0 ρΩ

0  ρΩ

I is an identity matrix with appropriate dimension.

Proof. Consider the following line-integral Lyapunov function candidate, ! Z t Z m m X X T i T i si ðI 1 ψ ÞX i  dψ þ si ð^x ðsÞÞRi x^ ðsÞ ds VðtÞ ¼ 2 x^ ðsÞ Γð0;ξÞ i ¼ 1

Z

þτM

t t  τM

t  τM

Z

t ν

T x^_ ðsÞ

m X

!

i¼1

si ð^x i ðsÞÞQi x^_ ðsÞ ds dν;

ð13Þ

i¼1

where X i 40, Ri 40, and Qi 40, i ¼ 1; 2; …; m, are positive matrices satisfying Eq. (10) to be determined later. The derivative of V(t) along the system (8) is given as follows, for t A ½t k h þ τk ; t kþ1 h þ τkþ1 Þ, ! m X T si ð^x i ðtÞÞX i ξðtÞ V_ ðtÞ ¼ ξ_ ðtÞ i¼1

Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

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m X

þξT ðtÞ

i¼1 m X

þ^x T ðtÞ Z  τM

i¼1 t

t  tM

!

m X

_ si ð^x i ðtÞÞX i ξðtÞ x^ T ðt  τM Þ ! si ð^x i ðtÞÞRi x^ ðtÞ þ τ2M x^_ ðtÞ T

x^_ ðsÞ T

m X

!

! si ð^x i ðt  τM ÞÞRi x^ ðt  τM Þ

i¼1 m X

!

si ð^x i ðtÞÞQi x^_ ðtÞ

i¼1

si ð^x i ðsÞÞQi x^_ ðsÞ ds:

ð14Þ

i¼1

Introducing the event-driven criterion Eq. (7), Eq. (14) can be simplified as, ! m X m X j T i _ V ðtÞ ¼ η ðtÞ si ðx ðtÞÞsj ð^x ðtÞÞΨ 1ij ηðtÞ i¼1j¼1

þτ2M ηT ðtÞ

m X

! sj ð^x

j¼1

j

ðtÞÞΨ T2j Qj Ψ 2j

Z ηðtÞ τM

t t  tM

m X

T x^_ ðsÞ

! sj ð^x ðtÞÞQj x^_ ðsÞ ds; j

j¼1

ð15Þ where ηT ðtÞ ¼ ½ξT ðtÞ x^ T ðt−τðtÞÞ eTk ðt−τðtÞÞ x^ T ðt−τM Þ; 3 2 Ψ 11ij X j B ij K −X j B ij K 0 7 6 ρΩ −ρΩ 0 7 6 n 7; Ψ 11ij ¼ symfX j Aij þ X j Lg þ I T Rj I 1 : 6 Ψ 1ij ¼ 6 1 n ðρ−1ÞΩ 0 7 5 4 n n n n −Rj Notice that Z t Z T −τM x^_ ðsÞQj x^_ ðsÞ ds ¼ −τM t−t M

t

T x^_ ðsÞQj x^_ ðsÞ ds−τM

t−τðtÞ

Z

t−τðtÞ

T x^_ ðsÞQj x^_ ðsÞ ds;

ð16Þ

t−τM

and according to Lemma 2, the following inequality holds: Z t Z t  τðtÞ Z t  τðtÞ T T  τM x^_ ðsÞQj x^_ ðsÞ ds r  x^_ ðsÞ dsQj x^_ ðsÞ ds Z

t  τM

Z

t  τM

Z

t  τM

t  τðtÞ

τðtÞ T T x^_ ðsÞ dsQj x^_ ðsÞ ds  x^_ ðsÞ dsQj τ  τðtÞ M t  τM t  τðtÞ  Z tt  τðtÞ Z t τðtÞ T  1 x^_ ðsÞ dsQj x^_ ðsÞ ds: τM  τðtÞ t  τðtÞ t  τðtÞ 

t

t

Z

t  τðtÞ

t  τM

x^_ ðsÞ ds ð17Þ

Further, according to Lemma 2, Eq. (17) is equivalent to Eqs. (18) and (19), Z t Z t  τðtÞ Z t  τðtÞ T T x^_ ðsÞQj x^_ ðsÞ ds r  2 x^_ ðsÞ dsQj x^_ ðsÞ ds  τM Z 

t  τM

t t  τðtÞ

T x^_ ðsÞ dsQj

Z

t  τM

t

t  τðtÞ

x^_ ðsÞ ds;

t  τM

ð18Þ

Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

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and

Z  τM Z 2

Z t  τðtÞ Z t  τðtÞ T T x^_ ðsÞQj x^_ ðsÞ dsr  x^_ ðsÞ dsQj x^_ ðsÞ ds t  τM t  τM t  τM Z t t T x^_ ðsÞ dsQj x^_ ðsÞ ds:

9

t

t  τðtÞ

ð19Þ

t  τðtÞ

Substitute Eqs. (18) and (19) into Eq. (15), respectively, " # m X m X j T i 2 T V_ ðtÞr η ðtÞ si ðx ðtÞÞsj ð^x ðtÞÞðΨ 1ij þ τM Ψ 2j Qj Ψ 2j þ Q1j Þ ηðtÞ;

ð20Þ

i¼1j¼1

and

" V_ ðtÞr ηT ðtÞ

#

m X m X i¼1j¼1

where

2

6 6 Q1j ¼ 6 6 4 2 6 6 Q2j ¼ 6 6 4

 I T1 Qj I 1

si ðxi ðtÞÞsj ð^x j ðtÞÞðΨ 1ij þ τ2M Ψ T2j Qj Ψ 2j þ Q2j Þ ηðtÞ;

n

I T1 Qj  3Qj

0 0

n

n

0

n

n

n

 2I T1 Qj I 1

2I T1 Qj

0

n

 3Qj

n

n

0 0

n

n

n

ð21Þ

3 0 7 2Qj 7 7; 0 7 5  2Qj 3 0 7 Qj 7 7: 0 7 5  Qj

By Schur complement and Lemma 3, if Eqs. (11) and (12) hold, for 8 i; j ¼ 1; 2; …; m, and s¼ 1,2, then V_ ðtÞo0. The observer gain matrix L and the controller gain matrix K can also be coobtained by the LMIs such that the system (8) is asymptotically stable. □ 3.2. Optimal robust event-driven controller design In this section, the robust event-driven controller is designed for the state-dependent nonlinear system with the external disturbance ωðtÞ. Consider the state-dependent nonlinear system with the external disturbance, x_ ðtÞ ¼ AðsðxðtÞÞÞxðtÞ þ BðsðxðtÞÞÞuðtÞ þ Bω ðsðxðtÞÞÞωðtÞ; zðtÞ ¼ CxðtÞ;

ð22Þ

ωðtÞA R is the external disturbance, and ωðtÞA L2 ½0; N. Bω ðsðxðtÞÞÞ ¼ i s ðx ðtÞÞB , B are known matrices with appropriate dimensions. i ωi ωi i¼1 Under the state observer (3) and the event-driven controller uðtÞ ¼ Kð^x ðt  τðtÞÞ ek ðt  τðtÞÞÞ, the augmented closed-loop system can be rewritten as,  m  X si ðxi ðtÞÞ  si ð^x i ðtÞÞ _ξðtÞ ¼ ðA i ξðtÞ þ B i Kð^x ðt  τðtÞÞ  ek ðt  τðtÞÞÞ þ B ωi ωðtÞÞ; 0 si ð^x i ðtÞÞ i¼1 ð23Þ rðtÞ ¼ CxðtÞ  Cx^ ðtÞ ¼ C1 ξðtÞ;

where Pm

m

Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

10

where



B ωi ¼

 Bωi ; 0

C 1 ¼ CI T2 ;

I2 ¼

  I : 0

Theorem 2. Considering the closed-loop system (23), for given constants γ, ρ, ρj, and matrix Ω, if there exist matrices X i 40, Ri 40, Qi 40, i ¼ 1; 2; …; m and matrix K, matrix L, such that 8 i; j ¼ 1; 2; …; m, and s ¼ 1,2, Eqs. (24) and (25) hold, then the system (23) with the controller u ¼ Kð^x ðt  τðtÞÞ  ek ðt  τðtÞÞÞ is robust asymptotically stable with the desired H 1 performance index γ: 3 2 T T T Θ1ij Ψ T2j Xj L X j B ωi 7 6 6 n τ  2 ðρ2 Q  2ρ IÞ 0 0 0 7 j M j j 7 6 7 6 ð24Þ 6 n n I 0 0 7o0; 7 6 7 6 n n n I 0 5 4 n n n n  γ2 I 2

Θ2ij

6 6 n 6 6 6 n 6 6 n 4

T

T

T

Ψ T2j

Xj

L

τM 2 ðρ2j Qj  2ρj IÞ n

0 I

0 0

0 0

n

n

I

n

n

n

0  γ2 I

n

X j B ωi

where

3 7 7 7 7 7o0; 7 7 5

ð25Þ

2

Θ1ij ¼ Φ1ij þ Q1j þ C; Θ2ij ¼ Φ1ij þ Q2j þ C;

CT1 C1 6 0 6 C ¼6 4 0 0

0 0

0 0

0

0

3 0 07 7 7: 05

0

0

0

Proof. Considering the integral function (13), the derivative of V(t) along the system (23) is similarly to Theorem 1, for any ωðtÞa 0, t A ½t k h þ τk ; t kþ1 h þ τkþ1 Þ, V_ ðtÞ þ r T ðtÞrðtÞ γ 2 ωT ðtÞωðtÞoηT ðtÞ½Ψ 1i þ τ2M Ψ T2j Qj Ψ 2j þ Q1j ηðtÞ T

T

þ ηT ðtÞX j B ωi ωðtÞ þ ωT ðtÞB ωi X j ηðtÞ þ ηT ðtÞCηðtÞ γ 2 ωT ðtÞωðtÞ;

ð26Þ

and V_ ðtÞ þ r T ðtÞrðtÞ γ 2 ωT ðtÞωðtÞoηT ðtÞ½Ψ 1i þ τ2M Ψ T2j Qj Ψ 2j þ Q2j ηðtÞ T

T

þ ηT ðtÞX j B ωi ωðtÞ þ ωT ðtÞB ωi X j ηðtÞ þ ηT ðtÞCηðtÞ γ 2 ωT ðtÞωðtÞ:

ð27Þ

By Schur complement and Lemma 3, if Eqs. (24) and (25) hold, from Eqs. (26) and (27), it can be concluded that ð28Þ V_ ðtÞr  r T ðtÞrðtÞ þ γ 2 ωT ðtÞωðtÞ: Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

11

Integrate both sides of Eq. (28) from t0 to t and letting t-1, we can get the following inequality with the zero initial condition: Z 1 Z 1 r T ðsÞrðsÞ dsr γ 2 ωT ðsÞωðsÞ ds; ð29Þ t0

t0

which implies that JrðtÞJ 2 r γ JωðtÞ J 2 . When ωðtÞ ¼ 0, according to Theorem 1, V_ ðtÞo0. The observer gain matrix L and the controller gain matrix K can also be obtained by the LMIs Eqs. (24) and (25) such that the system (23) is robust asymptotically stable, and the H 1 performance index is no larger than γ. □ Remark 3. From Theorem 2, we can know that if γ is known, this problem is suboptimal H 1 control problem. When γ is not given, we can get γ through “minγ”, that is the optimal H 1 problem.

Remark 4. In Theorems 1 and 2, one can see that for the given criterion parameter ρ40, by solving a set of LMIs, the event-driven matrix Ω, the observer gain matrix L and the controller gain matrix K all can be obtained. Also for the given event-driven gain matrix Ω, the criterion parameter ρ, the observer gain matrix L and the controller gain matrix K also can be obtained. Therefore, one can employ Theorems 1 and 2 to co-design the feedback gain matrix K, the observer gain matrix L and the event-driven parameters ρ, Ω.

Remark 5. Notice that the conditions Eqs. (24) and (25) (or Eqs. (11) and (12)) include information of the transmission delay τM. Thus, our method can be used to deal with the network transmission delay. For the given delay, by solving Eqs. (24) and (25) (or Eqs. (11) and (12)), the event-driven parameters Ω, ρ, the observer gain matrix L and the controller gain matrix K can be obtained. On the other hand, for given ρ40 and ρj, the upper bound of τM can be solved in terms of Eqs. (24) and (25) (or Eqs. (11) and (12)).

4. Numerical examples To demonstrate the effectiveness of Theorem 1, the following examples (Examples 1 and 2) are considered. Example 1. Considering the state-dependent nonlinear system in the form of (2), coefficient matrices are given as follows:      1:95 1:53 0 1 s1 ðx ðtÞÞ ¼ cos ðx1 ðtÞÞ; A1 ¼ ; B1 ¼ ; 1:71  2:35 1:32      2:83 1:53 0 2 1 s2 ðx ðtÞÞ ¼ 1  s1 ðx ðtÞÞ; A2 ¼ ; B2 ¼ : 1:71  2:27 1:53 The initial state is xð0Þ ¼ ½ 0:5 1:2T , the sampling period of the sensor is h¼ 0.2, the timedelay τðtÞ satisfies τðtÞr τ ¼ 0:06. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

12

Taking the appropriate parameter values of ρ, ρi and Ω, by Theorem 1, the Lyapunov function matrices with the form of Eq. (10) are obtained and given as follows: 2

2

3 0:0104  0:0558 7 7 7;  1:1739 5

 0:0915 0:3947

 0:0631  0:0063

 0:0063

1:6484

0:0104

 0:0558

 1:1739

1:7666

0:1618

 0:0915

 0:0631

0:0104

0:2555  0:0063

 0:0063 1:6820

0:1982 6  0:0915 6 X1 ¼ 6 4  0:0631

6  0:0915 6 X2 ¼ 6 4  0:0631

3

 0:0558 7 7 7;  1:1739 5

 0:0558  1:1739 1:7887     2:5312 2:4420  2:5312 R1 ¼ ; R2 ¼ ;  2:5312 2:8486  2:5312 2:8753     1:3471 0:1272 1:4659 0:1272 Q1 ¼ ; Q2 ¼ : 0:1272 1:3657 0:1272 1:5573 

0:0104 2:4785

The observer gain matrix L is obtained such that the estimated state x^ ðtÞ can converge asymptotically to the system state x(t), as shown in Fig. 3. Also the event driven controller gain matrix K is obtained to make the augmented closed-loop system is asymptotically stable which is depicted in Fig. 4, that is to say, the designed observerbased event-driven controller is effective to the system (1). The release instants and the release intervals of the event generator are shown in Fig. 5 to reflect the influence of the event generator on the release periods. It can be computed that our event generator leads to the average release period of 0.4063. Under the event generator, the average release period is significantly larger than the sampling period of the sensor (h¼ 0.2). As a result, the communication resources can be saved significantly, and the burden of the network communication can be reduced.

0.2

1.4

0.1

1.2

0

1

−0.1

0.8

−0.2

0.6

−0.3

0.4

−0.4

0.2

−0.5

0

5

10

15

20

0

0

5

10

15

20

Fig. 3. State observation error xðtÞ  x^ ðtÞ. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

13

1.4 1.2

x 2 (t)

1 0.8 0.6 0.4 0.2 0 −0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

x1 (t)

Fig. 4. State trajectory of the system with the observer-based event-driven controller.

1.4

Release intervals

1.2 1 0.8 0.6 0.4 0.2 0

0

5

10 Release instants (t/s)

15

20

Fig. 5. Release instants and release intervals with ρ ¼ 0:5.

Example 2. The model in this paper is widely used in practical applications. The tunnel diode circuit system [1] is taken as an example and described as, 1 1 V C ðt Þ V C ðt Þ þ iL ðt Þ; RL RD L_i L ðtÞ ¼  V C ðtÞ RE iL ðtÞ þ 1:5V C ðtÞ þ uðtÞ;

CV_ C ðtÞ ¼ 

Let x1 ðtÞ ¼ V C ðtÞ and x2 ðtÞ ¼ iL ðtÞ be the state variables. The system can be described in the form of (2) with the following parameters: " " # 0:01 1 #  RL1C  0:002 0 1 2 C  C m C s1 ðx1 ðt ÞÞ ¼ x1 ðt Þ; A1 ¼ RE ; B1 ¼ 1 ; 1 m  L L 2L " " # 1 0:002 1 #  RL C  C 0 1 C s2 ðx1 ðt ÞÞ ¼ 1  x21 ðt Þ; A2 ¼ ¼ ; B 2 1 ; 1 m  RLE L 2L where C ¼ 0:6F, L ¼ 0:7H, RE ¼ 2:7Ω and RL ¼ 1Ω. The initial state is xð0Þ ¼ ½ 0:4; 3:1T , m ¼ 43.1, and the sampling period is h¼ 0.1, the timedelay τðtÞ satisfies τðtÞr τ ¼ 0:07. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

14

According to Theorem 1, the Lyapunov function matrices satisfying Eq. (10) are obtained by Matlab LMIs toolbox and given as follows: 2 3 0:1030  0:0738 0:0079 0:0064 6  0:0738 1:1225  0:0600 0:0059 7 6 7 X1 ¼ 6 7; 4 0:0079  0:0600 1:1416  0:5579 5 2

0:0064

1:0919 6  0:0738 6 X2 ¼ 6 4 0:0079

0:0059

 0:5579

 0:0738 2:3115

0:0079  0:0600

3:1377

3 0:0064 0:0059 7 7 7;  0:5579 5

 0:0600 1:1584 0:0059  0:5579 3:1342     1:5037 1:1351  1:5037 R1 ¼ ; R2 ¼ ;  1:5037 5:0094  1:5037 4:9742     0:6356 0:0348 0:6027 0:0348 Q1 ¼ ; Q2 ¼ : 0:0348 0:5619 0:0348 0:5122 0:0064  1:0955

Similarly, the observer gain matrix L and the event driven controller gain matrix K are obtained, respectively. The state observer error trajectories and the closed-loop system state trajectory are given in Figs. 6 and 7, respectively, which can verify the feasibility of the proposed method. If selecting the sampling period h¼ 0.1, since τ ¼ 0:07, the corresponding allowable maximum transmission delay is 0.17. Considering the effect of the transmission delay, the average release period is obtained as 0.2774, the release instants and release interval of the system are shown in Fig. 8. The following examples (Examples 3 and 4) are given out to prove the effectiveness of Theorem 2. Example 3. Considering the state-dependent nonlinear system with the external disturbance, the coefficient matrices Ai and Bi are the same as Example 1, the coefficient matrices C and Bωi are 3.5

0.6 0.5

3

0.4

2.5

0.3 0.2

2

0.1

1.5

0 −0.1

1

−0.2

0.5

−0.3 −0.4

0

5

10

15

20

25

0

0

5

10

15

20

25

Fig. 6. State observation error xðtÞ  x^ ðtÞ. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

15

3.5 3

x 2 (t)

2.5 2 1.5 1 0.5 0 −0.4

−0.2

0

0.2

0.4

0.6

x1 (t)

Fig. 7. State trajectory of the closed-loop tunnel diode circuit system. 1.4

Release Intervals

1.2 1 0.8 0.6 0.4 0.2 0

0

5

10

15

20

25

Release Instants (t/s)

Fig. 8. Release instants and release intervals.

chosen as, 

C ¼ 0:35

 0 ;

 Bω1 ¼ Bω2 ¼

0 0:52

 :

The external disturbance is ( 0:23; 2r t r 5ðsÞ ωðtÞ ¼ 0; otherwise: According to Theorem 2, the Lyapunov function matrices satisfying Eq. (10) are obtained by Matlab LMIs toolbox and given as follows: 2 3 0:1608  0:0316  0:0307  0:0007 6  0:0316 0:1251  0:0042  0:0103 7 6 7 X1 ¼ 6 7; 4  0:0307  0:0042 2:0011  1:3986 5  0:0007

 0:0103

 1:3986

2:0276

Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

16

0.2

1.4

0.1

1.2

0

1

−0.1

0.8

−0.2

0.6

−0.3

0.4

−0.4

0.2

−0.5

0

5

10

15

20

25

30

0

0

5

10

15

20

25

30

Fig. 9. State observation error xðtÞ  x^ ðtÞ.

1.4 1.2

x2 (t)

1 0.8 0.6 0.4 0.2 0 −0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

x1 (t)

Fig. 10. State trajectory of the system with the observer-based event-driven controller.

2

0:1870 6  0:0316 6 X2 ¼ 6 4  0:0307

 0:0316 0:1258

 0:0307  0:0042

 0:0042

1:9740

3  0:0007  0:0103 7 7 7;  1:3986 5

 0:0007  0:0103  1:3986 2:0503     3:1549  3:1223 2:7688  3:1223 R1 ¼ ; R2 ¼ ;  3:1223 3:3494  3:1223 3:5219     1:3009  0:1051 0:9613  0:1051 Q1 ¼ ; Q2 ¼ :  0:1051 1:3065  0:1051 1:2157 The other conditions are all the same as Example 1. According to Theorem 2, the observer gain matrix L is obviously obtained such that the estimated state x^ ðtÞ can converge asymptotically to the system state x(t), as shown in Fig. 9. Also the event driven controller gain matrix K is obtained to make the augmented closed-loop system is robust asymptotically stable with the H 1 performance index γ ¼ 0:3738, as shown in Fig. 10. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

17

Release intervals

2.5 2 1.5 1 0.5 0

0

5

10

15

20

25

30

Release instants (t/s)

Fig. 11. Release instants and release intervals with ρ ¼ 0:4. Table 1 Under the different ρ, the average release periods and the H 1 performance index γ (unit: s). ρ

0

0.4

0.7

0.9

The average release period The H 1 performance index γ

0.2600 0.3946

0.4686 0.3738

0.5521 0.3739

0.6160 0.3696

Moreover, the release instants and the release intervals of the event generator are shown in Fig. 11. In addition, a more detailed comparison for different cases are shown in Table 1. From Table 1, it is easy to know that the larger ρ ðρ A ½0; 1ÞÞ, the larger average release period and the better robustness of the system. Example 4. The tunnel diode circuit system [2] is taken as a model-plant mismatched example. The difference is that we consider the tunnel diode circuit system with the input vector u(t). Let x1 ðtÞ ¼ V C ðtÞ and x2 ðtÞ ¼ iL ðtÞ be the state variables. The circuit is governed by the following state equations: Cx_ 1 ðtÞ ¼  0:002x1 ðtÞ 0:01x31 ðtÞ þ x2 ðtÞ; L_x 2 ðtÞ ¼  x1 ðtÞ  10x2 ðtÞ þ uðtÞ þ ωðtÞ; zðtÞ ¼ x1 ðtÞ; where C ¼ 20 mF, L ¼ 1 H. Assuming that Jx1 ðtÞ J r3, the nonlinear network system can be approximated by the following state-dependent nonlinear system in the form of (2) with the following parameters: 8 3þx1 > < 3 ;  3 r x1 r 0; s1 ðx1 ðt ÞÞ ¼ 3 3 x1 ; 0 r x1 r 3; ; s2 ðx1 ðt ÞÞ ¼ 1 s1 ðx1 ðt ÞÞ; > : 0; otherwise:      0:1 50  4:6 50 A1 ¼ ; A2 ¼ ;  1  10  1  10 Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

18

0.5

0.1

0

−0.1

0

−0.2 −0.5

−0.3 −0.4

−1

−0.5 −0.6

−1.5

−0.7 −2

0

2

4

6

8

10

−0.8

0

2

4

6

8

10

Fig. 12. State observation error xðtÞ  x^ ðtÞ.

0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −2

−1.5

−1

−0.5

0

0.5

Fig. 13. State trajectory of the closed-loop tunnel diode circuit system.

0.9

Releae Intervals

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

2

4

6

8

10

Release Instants (t/s)

Fig. 14. Release instants and release intervals.

Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

J. Feng, N. Li / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

B1 ¼ B2 ¼

  0 1

;

Bω1 ¼ Bω2 ¼

19

  0 : 1

The initial state is xð0Þ ¼ ½0:2;  0:8T , the sampling period of the sensor is h¼ 0.1, the timedelay τðtÞ satisfies τðtÞr τ ¼ 0:07. According to Theorem 2, the Lyapunov function matrices with the form (Eq. (10)) are obtained by Matlab LMIs toolbox and given as follows: 2

 0:0001

10:7827 0:4018

0:4018 2:2641

 0:4443 7 7 7; 5:1063 5

0:3467

 0:4443

5:1063

103:7769

0:2047

0:9107

 0:0001

0:3467

10:2373 0:4018

0:4018 1:9946

6 0:9107 6 X1 ¼ 6 4  0:0001 2

3

0:9107

0:2232

6 0:9107 6 X2 ¼ 6 4  0:0001

0:3467

3

 0:4443 7 7 7; 5:1063 5

0:3467  0:4443 5:1063 106:4611    0:0547 0:0306 3:9845 0:0306 R1 ¼ ; R2 ¼ ; 0:0306 19:8974 0:0306 144:8611     0:3781 0:0009 0:4580 0:0009 Q1 ¼ ; Q2 ¼ : 0:0009 0:4219 0:0009 0:4733 

Similarly, the observer gain matrix L and the controller gain matrix K are obtained, respectively. The state observer error trajectories and the closed-loop system state trajectory are given in Figs. 12 and 13, respectively. From Fig. 13, it is easy to know that the system is asymptotically stable with the H 1 performance index γ ¼ 1:0902, which can verify the feasibility of the proposed method. In order to reflect the influence of the event generator, the release instants and the release intervals of the event generator are depicted in Fig. 14. By simple calculation, the average period is 0.3085, which is also larger than the sampling period (0.1).

5. Conclusions For the state-dependent nonlinear system, both the observer-based event-driven feedback control and the robust observer-based event-driven feedback control have been investigated in this paper. The event-driven criterion was proposed to determine whether the sampling signals should be sent to the controller. Based on the designed state-dependent observer, the augmented closed-loop system with the event-driven state feedback controller was rewritten as the delayed system. The integral Lyapunov function candidate was introduced to reduce the conservatism of asymptotic stability conditions. The state observer gain matrix L, the controller gain matrix K and the event-driven parameters ρ and Ω were obtained in the terms of solutions to a set of LMIs such that the closed-loop system was asymptotically stable with the H 1 performance index γ. The effectiveness of the proposed method was illustrated by the numerical examples and the tunnel diode circuit systems. Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012

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Please cite this article as: J. Feng, N. Li, Design of observer-based event-driven controllers for a class of statedependent nonlinear systems, Journal of the Franklin Institute. (2016), http://dx.doi.org/10.1016/j. jfranklin.2016.02.012